A left invariant vector field is a type of vector field that remains unchanged under a left action by a group of transformations. This means that the vector field's direction and magnitude are preserved when the group's transformation is applied to it.
A gauge transformation is a type of transformation used in physics to describe the same physical system in different ways. It is often used in the context of gauge theories, such as electromagnetism, where different gauge transformations result in the same physical phenomenon.
A gauge transformation applied to a left invariant vector field results in a new left invariant vector field. This means that the direction and magnitude of the vector field are preserved, but it may be defined on a different coordinate system due to the transformation.
Left invariant vector fields are important in gauge theories because they represent physical quantities that are preserved under a gauge transformation. This allows for a consistent description of physical systems despite changes in the coordinate system.
Yes, a left invariant vector field can be transformed by other types of transformations, such as right actions or general coordinate transformations. However, under a gauge transformation, the vector field remains left invariant and its physical significance is preserved.